Global liftings between inner forms of \(\mathrm{GSp}(4)\) (Q6585658)

Let \(H\) be a quasi-split group over \(\mathbb{Q}\) such that \(H(\mathbb{R})\) admits discrete series representations, and let \(G\) be an inner form of \(G\) such that \(G(\mathbb{R})\) is anisotropic. Let \(\pi\) be an irreducible automorphic representation \(\pi\) of \(G(\mathbb{A})\) which is neither weakly \(H\)-Eisenstein nor weakly endoscopic. Applying the cohomological trace formula, it is proved in this paper (Theorem7.1) that \(\pi\) admits a weak global lifting to \(H(\mathbb{A})\). For \(H=\mathrm{GSp}(4)\) finer questions such as the local lifting at ramified places and the image of the lifting are addressed. In particular, the authors prove the recent conjectures of Ibukiyama and Kitayama on paramodular newforms of square-free level.